Combinatorial and stochastic properties of ranked tree-child networks

TL;DR

This paper introduces ranked tree-child networks, a mathematically tractable subclass of phylogenetic networks, and provides combinatorial enumeration, probabilistic analysis, and conjectures on their properties, with potential applications to other network classes.

Contribution

It demonstrates how adding a ranking structure makes tree-child networks combinatorially manageable and explores their enumeration, sampling, and probabilistic properties.

Findings

Explicit formulas for counting ranked tree-child networks

Analysis of random walk lengths and cherry distributions

Conjecture on the scaling limit of lineage counting process

Abstract

Tree-child networks are a recently-described class of directed acyclic graphs that have risen to prominence in phylogenetics (the study of evolutionary trees and networks). Although these networks have a number of attractive mathematical properties, many combinatorial questions concerning them remain intractable. In this paper, we show that endowing these networks with a biologically relevant ranking structure yields mathematically tractable objects, which we term ranked tree-child networks (RTCNs). We explain how to derive exact and explicit combinatorial results concerning the enumeration and generation of these networks. We also explore probabilistic questions concerning the properties of RTCNs when they are sampled uniformly at random. These questions include the lengths of random walks between the root and leaves (both from the root to the leaves and from a leaf to the root); the distribution of the number of cherries in the network; and sampling RTCNs conditional on displaying a given tree. We also formulate a conjecture regarding the scaling limit of the process that counts the number of lineages in the ancestry of a leaf. The main idea in this paper, namely using ranking as a way to achieve combinatorial tractability, may also extend to other classes of networks.